# Canonical transformation to diagonalize Bosonic Hamiltonian

The Hamiltonian of the system of bosons ($$a$$, $$a^{\dagger}$$, $$b^{\dagger}$$ & $$b$$ are Bose operators) is: $$$$H=\epsilon_{1} a^{\dagger}a+\epsilon_{2}b^{\dagger}b+\frac{\Delta}{2}\left(a^{\dagger}b^{\dagger}+ba \right)$$$$

where $$\epsilon_{1}$$, $$\epsilon_{2}$$, and $${\Delta}$$ are real and positive, $${\Delta}$$ < ($$\epsilon_{1}$$ + $$\epsilon_{2}$$). I am trying to find a Canonical Transformation to diagonalize this Hamiltonian. And afterward to find expressions for the eigenenergies and parameters of the transformation. I am not sure whether first I need to switch to any other space like momentum etc and using Bogoliubov Transformation. Any help and hint will be highly appreciated.

You need to make sure the bosonic commutation realtions hold for any basis you choose. For that you need the equivalent of a $$z$$-Pauli matrix $$\sigma_3 = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix},$$ where this is just an example for two bosonic operators, the size of the matrix is 2 x #bosons.
You start by expanding your Hamiltonian to a Nambu space Hamiltonian, i.e. in momentum space this basis would be $$\begin{pmatrix} a_k & b_k & a^\dagger_{-k} & b^\dagger_{-k} \end{pmatrix}$$. Now that you have your Hamiltonian written in this basis, you diagonalize the matrix $$\sigma_3 H$$.
You could represent $$\hat{H}$$ as $$$$\hat{H} = \begin{pmatrix}\hat{a}^{\dagger}&\hat{b}^{\dagger}&\hat{b}\end{pmatrix}\underbrace{\begin{pmatrix} \epsilon_{1} & 0&\frac{\Delta}{2}\\ 0&\epsilon_2&0\\ \frac{\Delta}{2}&0&0 \end{pmatrix}}_{\equiv M} \begin{pmatrix}\hat{a}\\\hat{b}\\\hat{b}^{\dagger}\end{pmatrix}$$$$
The only thing left to do is to calculate the Eigenvalues and Eigenvectors of $$M$$